Gravity-induced viscous flow around a cylinder

[member 428835]

Homework Statement

A viscous liquid with density and viscosity $\rho$ and $\mu$ respectively is discharged onto the upper surface of a cylinder with radius $a$ at a volume flow rate $Q$. This is a gravity-driven flow, and it forms a film around the cylinder--see picture.

What is the thickness $h$ of the layer as a function $\theta$, $a$, $\rho$, $\mu$, $g$, and $Q$.

Homework Equations

Navier-Stokes
Continuity

The Attempt at a Solution

First off, let's assume the flow is incompressible and viscous-dominated, Newtonian, steady, 2-D, and that no pressure gradient is present. Then continuity and Navier-Stokes equations are $$\nabla \cdot \vec{V} = 0$$ and $$g\hat{j} = \nu \nabla^2 \vec{V}$$
Now if we adopt a cylindrical coordinate system where $x=rcos\theta$ and $y=r\sin\theta$ we have
$$\frac{1}{r}\frac{\partial (r u_r)}{\partial r}+\frac{1}{r}\frac{\partial u_\theta}{\partial \theta}=0$$ and
$$g(\sin\theta \hat{e_r}+\cos\theta \hat{e_\theta})=\nu \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial \vec{V} }{\partial r} \right)+\frac{1}{r^2} \frac{\partial^2 \vec{V}}{\partial \theta^2}$$

where $\vec{V} = u_\theta \hat{e_\theta}+u_r\hat{e_r}$. Boundary conditions would be no slip along the surface, $\vec{V} = 0$ at $r=a$. Another would be no stress along the surface of the thin film. And lastly we would have some incoming velocity related to $Q$, which would be the velocity at $r=a,\theta=0$. Any ideas how to proceed?

Thanks so much!

 

[Chestermiller]

I would start out by looking at flow down an inclined plane of constant angle, and seeing where that takes me. It's got to be a pretty good approximation for most angles on the cylinder.

Chet

 

[member 428835]

Thanks for the response Chet! I did this but how can you extrapolate this to a cylinder?

 

[Nidum]

A Google search turned up useful information about this problem .

Search on ' viscous thin film flow around a cylinder '

 

[Chestermiller]

Thanks for the response Chet! I did this but how can you extrapolate this to a cylinder?

If the thickness is varying very gradually with theta, it should give an accurate result locally, except for at the leading and trailing edges.